It follows from [1]} (see also [2]}) that there exists a constant \(c > 0\)  such that
\(\mathbb {P}\left(\max _{\tau ^O_{k^*} < t < \tau ^O_{{k^*} + 1}} \left\lbrace  (t - \tau ^O_{k^*}) \left(\overline{\varepsilon }^O_{\tau ^O_{k^*} :t}\right)^2 + ( \tau ^O_{{k^*} + 1} - t) \left(\overline{\varepsilon }^O_{t :\tau ^O_{{k^*} + 1}}\right)^2 \right\rbrace  > c\, \underline{\sigma }^2 \log \log \overline{\lambda }\right) \rightarrow 1.\) 
